The non-minimal coupling term changes the physical landscape significantly, in a qualitative fashion. This is related to the dual role that it plays: it acts as part of the potential for the scalar field but also contributes to the Einstein-Hilbert term for gravity. As a consequence, the old parameters when \(\xi =0\)  such as the VEV \(v\) , cosmological constant \(\Lambda \)  and \(\alpha \)  (proportional to the inverse of Newton's constant) become effectively the VEV \(v_{eff}\) , the asymptotic cosmological constant \(\Lambda _{eff}\)  and \(\alpha _{eff}\)  respectively that now depend on the coupling \(\xi \) . The novel feature that emerges is that in an AdS\(_3\)  background, where \(\Lambda _{eff}\)  is non-zero and negative, there exists a critical coupling \(\xi _c\)  where the VEV \(v_{eff}\)  is zero for \(\xi \)  at or above \(\xi _c\)  but is non-zero when \(\xi \)  crosses below \(\xi _c\) . When the VEV crosses from zero to non-zero at \(\xi _c\) , the local \(U(1)\)  gauge symmetry is spontaneously broken corresponding to a phase transition to a vortex. The critical coupling \(\xi _c\)  acts like the analog of the critical temperature \(T_c\)  in Ginzburg-Landau (GL) mean-field theory where the order parameter is zero above \(T_c\)  but is non-zero below \(T_c\)  [1]}, [2]}. There is a second-order phase transition when the temperature crosses below \(T_c\)  and this is typically accompanied by a symmetry that is spontaneously broken. The analogy between \(\xi _c\)  and \(T_c\)  can be made quantitative. Near \(\xi _c\) , we show that the VEV \(v_{eff}\)  has a power-law behaviour proportional to \(|\xi -\xi _c|^{1/2}\)  which is similar to the \(|T-T_c|^{1/2}\)  power-law behaviour of the order parameter near \(T_c\)  in GL mean-field theory [1]}, [2]}; both have a critical exponent of \(1/2\) . The plot of the VEV versus the coupling \(\xi \)  looks very similar to the plot of the order parameter versus temperature \(T\)  in GL mean-field theory and in both cases there is a discontinuity in the slope at the critical point where the slope diverges.
